A Composition Theorem for Conical Juntas

Transcription

1 A Composition Theorem for Conical Juntas Mika Göös T.S. Jayram University of Toronto IBM Almaden Göös and Jayram A composition theorem for conical juntas 29th May / 12

2 Motivation Randomised communication AND-OR trees 1 Set-disjointness ORn AND 2 2 Tribes AND n OR n AND 2 Ω(n) Ω(n) [KS 87] [Raz 91] [BJKS 04] (info) [JKS 03] (info) [HJ 13] k AND n 1/k OR n 1/k AND 2 n/2 O(k) [JKR 09] (info) [LS 10] (info) Göös and Jayram A composition theorem for conical juntas 29th May / 12

3 Motivation Randomised communication AND-OR trees 1 Set-disjointness ORn AND 2 2 Tribes AND n OR n AND 2 Ω(n) Ω(n) [KS 87] [Raz 91] [BJKS 04] (info) [JKS 03] (info) [HJ 13] k AND n 1/k OR n 1/k AND 2 n/2 O(k) [JKR 09] (info) [LS 10] (info) log n AND 2 OR 2 AND 2 O(n ) Ω( n) Gap! [Snir 85] [JKZ 10] Göös and Jayram A composition theorem for conical juntas 29th May / 12

4 New tool Conical juntas Communication-to-query theorem [GLMWZ 15]: For every boolean function f : {0, 1} n {0, 1}, BPP cc ɛ ( f IP log n ) Ω(deg + ɛ ( f )) f Compose with IP f z 1 z 2 z 3 z 4 z 5 IP IP IP IP IP x 1 y 1 x 2 y 2 x 3 y 3 x 4 y 4 x 5 y 5 Göös and Jayram A composition theorem for conical juntas 29th May / 12

5 New tool Conical juntas Communication-to-query theorem [GLMWZ 15]: For every boolean function f : {0, 1} n {0, 1}, BPP cc ɛ ( f IP log n ) Ω(deg + ɛ ( f )) Conical juntas: Nonnegative combination of conjunctions OR 2 : 1 2 x x x 1x x 1 x 2 Approximate conical junta degree deg + ɛ ( f ) is the least degree of a conical junta h such that x {0, 1} n : f (x) h(x) ɛ. Göös and Jayram A composition theorem for conical juntas 29th May / 12

6 New tool Conical juntas Communication-to-query theorem [GLMWZ 15]: For every boolean function f : {0, 1} n {0, 1}, BPP cc ɛ ( f IP log n ) Ω(deg + ɛ ( f )) Conical juntas: Nonnegative combination of conjunctions OR 2 : 1 2 x x x 1x x 1 x 2 Approximate conical junta degree deg + ɛ ( f ) is the least degree of a conical junta h such that x {0, 1} n : f (x) h(x) ɛ. Previous talk: 0-1 coefficients = Unambiguous DNFs Göös and Jayram A composition theorem for conical juntas 29th May / 12

7 New tool Big picture BPP dt WAPP dt approx. junta deg Open problem Classical world [GLMWZ 15] BPP cc WAPP cc approx. rank + BQP dt AWPP dt approx. poly deg Open problem Quantum world [She 09, SZ 09] BQP cc AWPP cc approx. rank Göös and Jayram A composition theorem for conical juntas 29th May / 12

8 This work: A Composition Theorem for Conical Juntas That is: Want to understand approximate conical junta degree of f g in terms of f and g Göös and Jayram A composition theorem for conical juntas 29th May / 12

9 Our results Applications = M M M M M M M M M M M M M NAND 4 (OR AND) 2 MAJ 3 3 Query: deg + 1/n (NAND k ) Ω(n ) deg + 1/n (MAJ k 3 ) Ω( k ) Göös and Jayram A composition theorem for conical juntas 29th May / 12

10 Our results Applications = M M M M M M M M M M M M M NAND 4 (OR AND) 2 MAJ 3 3 Query: deg + 1/n (NAND k ) Ω(n ) deg + 1/n (MAJ k 3 ) Ω( k ) Previously: Note: O(2.65 k ) BPP dt (MAJ k 3 ) Ω(2.57 k ) [JKS 03, LNPV 06, Leo 13, MNSSTX 15] Unamplifiability of error Göös and Jayram A composition theorem for conical juntas 29th May / 12

11 Our results Applications = M M M M M M M M M M M M M NAND 4 (OR AND) 2 MAJ 3 3 Query: deg + 1/n (NAND k ) Ω(n ) deg + 1/n (MAJ k 3 ) Ω( k ) Communication: BPP cc (NAND k ) Ω(n ) BPP cc (MAJ k 3 ) Ω(2.59 k ) Note: Log-factor loss Göös and Jayram A composition theorem for conical juntas 29th May / 12

12 Formalising the composition theorem Average degree For h = w C C, adeg x (h) := w C C C(x) adeg(h) := max x adeg x (h) Göös and Jayram A composition theorem for conical juntas 29th May / 12

13 Formalising the composition theorem Average degree For h = w C C, adeg x (h) := w C C C(x) adeg(h) := max x adeg x (h) Simplification: Consider zero-error conical juntas Example: adeg(or 2 ) = 3/2, adeg(maj 3 ) = 8/3 Göös and Jayram A composition theorem for conical juntas 29th May / 12

14 Formalising the composition theorem Average degree For h = w C C, adeg x (h) := w C C C(x) adeg(h) := max x adeg x (h) Simplification: Consider zero-error conical juntas Example: adeg(or 2 ) = 3/2, adeg(maj 3 ) = 8/3 First formalisation attempt adeg( f g) adeg( f ) min { adeg(g), adeg( g) }? Göös and Jayram A composition theorem for conical juntas 29th May / 12

15 Formalising the composition theorem Average degree For h = w C C, adeg x (h) := w C C C(x) adeg(h) := max x adeg x (h) Simplification: Consider zero-error conical juntas Example: adeg(or 2 ) = 3/2, adeg(maj 3 ) = 8/3 First formalisation attempt adeg( f g) adeg( f ) min { adeg(g), adeg( g) }? Counter-example! adeg(or 2 MAJ 3 ) = < 4 Göös and Jayram A composition theorem for conical juntas 29th May / 12

16 Formalisation LP duality cf. [She 13, BT 15] adeg(h; b 0, b 1 ) charge bi for reading an input bit that is i Primal / Dual for average degree of f min adeg ( ) w C C; b 0, b 1 subject to w C C(x) = f (x), x w C 0, C max Ψ, f subject to Ψ, C adeg(c; b 0, b 1 ), C Ψ(x) R, x Göös and Jayram A composition theorem for conical juntas 29th May / 12

17 Formalisation Statement of theorem Regular certificates: Circumventing the counter-example Require that Ψ is balanced (has a primal meaning!) Göös and Jayram A composition theorem for conical juntas 29th May / 12

18 Formalisation Statement of theorem Regular certificates: Circumventing the counter-example Require that Ψ is balanced (has a primal meaning!) Require that Ψ 1 and Ψ 0 for f and f share structure Göös and Jayram A composition theorem for conical juntas 29th May / 12

19 Formalisation Statement of theorem Regular certificates: Circumventing the counter-example Require that Ψ is balanced (has a primal meaning!) Require that Ψ 1 and Ψ 0 for f and f share structure Composition Theorem Suppose we have regular LP certificates witnessing adeg(g) b 1 adeg( f ; b 0, b 1 ) a 1 adeg( g) b 0 adeg( f ; b 0, b 1 ) a 0 then f g admits a regular LP certificate witnessing adeg( f g) a 1 adeg( f g) a 0 Göös and Jayram A composition theorem for conical juntas 29th May / 12

20 Regular certificates for MAJ 3 0 1/2 1/2 5/6 5/6 5/6 5/6 5/6 5/6 1/6 1/6 1/6 + = /6 5/6 5/6 0 Ψ 1 0 ˆΨ 1 0 Ψ 1 + ˆΨ 1 MAJ 3 MAJ3 2 MAJ3 3 MAJ3 4 # dual variables: lower bound: Open! Göös and Jayram A composition theorem for conical juntas 29th May / 12

21 Subsequent application Eight-author paper: Anshu, Belovs, Ben-David, Göös, Jain, Kothari, Lee, and Santha [ECCC 16] 1 total F : BPP cc (F) Ω(BQP cc (F) 2.5 ) 2 total F : BPP cc (F) log 2 o(1) χ(f) Göös and Jayram A composition theorem for conical juntas 29th May / 12

22 Subsequent application Eight-author paper: Anshu, Belovs, Ben-David, Göös, Jain, Kothari, Lee, and Santha [ECCC 16] 1 total F : BPP cc (F) Ω(BQP cc (F) 2.5 ) 2 total F : BPP cc (F) log 2 o(1) χ(f) Proof idea for 1 deg + ɛ (SIMON n AND n OR n ) Ω(n 2.5 ) Communication-to-query BPP cc (SIMON n AND n OR n IP log n ) Ω(n 2.5 ) Cheat sheet technique BPP cc( (SIMON n AND n OR n IP log n ) CS ) Ω(n 2.5 ) Göös and Jayram A composition theorem for conical juntas 29th May / 12

23 Open problems Composition theorems: Explain why our composition theorem works!-) Better certificates for MAJ3 k? Does a composition theorem hold for BPP dt? Simulation theorems: Communication-to-query simulation for BPP? Constant-size gadgets for junta-based simulation? More things to do with conical juntas? Göös and Jayram A composition theorem for conical juntas 29th May / 12

24 Open problems Composition theorems: Explain why our composition theorem works!-) Better certificates for MAJ3 k? Does a composition theorem hold for BPP dt? Simulation theorems: Communication-to-query simulation for BPP? Constant-size gadgets for junta-based simulation? More things to do with conical juntas? Cheers! Göös and Jayram A composition theorem for conical juntas 29th May / 12